Multiple-input multiple-output (MIMO) communication systems provide a spectrally-efficient means for communicating over wireless channels. Such systems can be classified into coherent systems in which the receiver has access to reliable channel state information (CSI) and non-coherent systems in which this information is not available at either the transmitter or the receiver. Coherent systems are more straightforward to design than their non-coherent counterparts. However, the analysis of coherent systems usually does not account for the cost of the resources that have to be expended to acquire reliable CSI. Ignoring this cost in static and slow fading scenarios is generally tolerable. Ignoring it in fast fading scenarios, which arise in wireless systems with high mobility, can be rather misleading. A mobile device that is traveling at 60 miles per hour has a fading coherence of about 3 ms. If the mobile system is operated at 1.9 GHz and has a symbol rate of 30K symbols per second, 3 ms allows for 50-100 symbol periods. If several training symbols per antenna are needed, only a few antennas may be trained during a fading coherence period.
In order to alleviate the cost of acquiring reliable CSI in fast fading scenarios, it is often believed to be desirable to use non-coherent signaling strategies, that is, strategies that do not require the receiver to have access to CSI. It has been shown that, for spatially-white MIMO channels with independent identically distributed (i.i.d.) block Rayleigh fading coefficients, input matrices that achieve the ergodic capacity can be expressed in the form of an isotropically distributed unitary component and a diagonal component with non-negative entries (Hochwald and Marzetta, “Unitary space-time modulation multiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 46, pp. 543-564, March 2000, which is hereby incorporated herein in its entirety by reference). A closed form expression for the asymptotically high signal-to-noise ratio (SNR) ergodic capacity of this type of channel has been determined for the cases where the coherence time of the channel exceeds and is below a certain threshold. When the coherence time of the channel exceeds the threshold, it was shown that, for sufficiently high SNRs, it is optimal, from a capacity perspective, for the diagonal component to be a scaled identity matrix and for the unitary matrix to be isotropically distributed on the so-called Grassmann manifold and to have the number of antennas transmitting less than or equal to half the number of symbol durations spanned by the coherence interval of the channel. Various techniques for designing rate-efficient Grassmannian constellations have been shown (L. Zheng and D. N. C. Tse, “Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel,” IEEE Trans. Inf. Theory, vol. 48, pp. 359-383, February 2002, which is hereby incorporated herein in its entirety by reference). For the case where the SNR is below the threshold, the unitary component was shown to have the same distribution as when above the threshold, but the optimal distribution for the diagonal component was shown to be that of the square root of the eigenvalues of a beta-distributed random matrix. In contrast with the high SNR cases, for low SNRs, it has been shown that the optimal diagonal component has at most one non-zero entry at any given time.
The proximity of antennas to each other in practical wireless systems renders the realization of spatially-white channels generally difficult, even when the distance between antennas exceeds multiple wavelengths. This proximity induces correlation between the random entries of the channel matrix at the transmitter and receiver sides. A convenient means for characterizing this correlation mathematically is the so-called Kronecker model, whereby the spatially-correlated channel matrix is represented by left and right multiplication of a spatially-white channel matrix with transmitter and receiver covariance matrices, respectively. Fortunately, these matrices are dominated by the locations, geometries and beam patterns of the antennas at the receiver and the transmitter. Being characteristic of the wireless devices on which the antennas are mounted, these matrices can be accurately estimated and made available to the transmitter and the receiver. The impact of these matrices on the proper signaling methodology and the achievable rate can be quite significant. For the case in which the correlation between the channel coefficients is described by the Kronecker model, it has been shown that for spatially white channels, increasing the number of transmit antennas is almost surely beneficial. It has also been shown that the optimal input covariance can be expressed as the product of an isotropically distributed unitary component, a diagonal component with non-negative entries and a deterministic component comprising the eigenvectors of the transmitter covariance matrix.